Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Topic Video
Question
please help
![### Geometry Problem: Rhombus Angle Measurement
**Question:**
What is the measure of ∠1 in rhombus ABCD?
**Diagram Description:**
In the given diagram, there is a rhombus labeled ABCD. The diagonals of the rhombus intersect at point O, forming four angles at the center. The diagonals bisect each other, and the properties of a rhombus dictate that these diagonals split the rhombus's angles into two equal halves.
The labeling of the angles around point O is provided as follows:
- ∠AOB is labeled as 1
- ∠AOD is labeled as 2
- ∠COD is labeled as 3
- ∠BOC is labeled as 4
Additionally, one of the angles, ∠ADC, is provided as 53°.
**Multiple Choice Options:**
- 53°
- 90°
- 106°
- 180°
**Explanation of the Problem:**
In a rhombus, the diagonals bisect the angles at the vertices and are also perpendicular bisectors of each other. Given that ∠ADC is 53°, the diagonal AD bisects this angle, forming two equal angles at point O.
Since the diagonals of a rhombus bisect the internal angles, each part of the bisected angle is half of the original. Therefore, ∠AOD and ∠COD each are:
\[
\angle AOD = \angle COD = \frac{53°}{2} = 26.5°
\]
It's a property of the rhombus's diagonals to meet at right angles (90°). Thus, ∠AOB, labeled as ∠1, being one of the angles formed by the intersection of the diagonals, should be 90°.
**Answer:**
The measure of ∠1 is **90°**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47897a37-2eac-43d5-a731-3aff060a1d87%2Fe42511c7-b92a-4c5c-a2a7-8a5d729b5600%2Fpa5dcgl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Rhombus Angle Measurement
**Question:**
What is the measure of ∠1 in rhombus ABCD?
**Diagram Description:**
In the given diagram, there is a rhombus labeled ABCD. The diagonals of the rhombus intersect at point O, forming four angles at the center. The diagonals bisect each other, and the properties of a rhombus dictate that these diagonals split the rhombus's angles into two equal halves.
The labeling of the angles around point O is provided as follows:
- ∠AOB is labeled as 1
- ∠AOD is labeled as 2
- ∠COD is labeled as 3
- ∠BOC is labeled as 4
Additionally, one of the angles, ∠ADC, is provided as 53°.
**Multiple Choice Options:**
- 53°
- 90°
- 106°
- 180°
**Explanation of the Problem:**
In a rhombus, the diagonals bisect the angles at the vertices and are also perpendicular bisectors of each other. Given that ∠ADC is 53°, the diagonal AD bisects this angle, forming two equal angles at point O.
Since the diagonals of a rhombus bisect the internal angles, each part of the bisected angle is half of the original. Therefore, ∠AOD and ∠COD each are:
\[
\angle AOD = \angle COD = \frac{53°}{2} = 26.5°
\]
It's a property of the rhombus's diagonals to meet at right angles (90°). Thus, ∠AOB, labeled as ∠1, being one of the angles formed by the intersection of the diagonals, should be 90°.
**Answer:**
The measure of ∠1 is **90°**.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning